Hi I need help with this stat question

Question

Improving SAT scores. Refer to the Chance (Winter 2001) study of students who paid a private tutor (or coach) to help them improve their Standardized Assessment Test (SAT) scores, Exercise 4.50 (p. 223). Recall that multiple regression was used to estimate the effect of coaching on SAT–Mathematics scores, where

y = SAT–Math score

x1 = score on PSAT

x2 = {1 if student was coached, 0 if not}

(a) Write a complete second-order model for E(y) as a function of x1 and x2.

(b) Give the equation of the curve relating E(y) to x1 for noncoached students. Identify the y-intercept, shift parameter, and rate of curvature in the equation.

(c) Repeat part b for students who have been coached on the SAT.

(d) How would you test to determine if coaching has an effect on SAT–Math scores?

Answer 1

(a) A complete second-order model for E(y) as a function of x1 and x2 is given by:

E(y) = β0 + β1x1 + β2x2 + β3x1^2 + β4x2^2 + β5x1x2

where β0, β1, β2, β3, β4, and β5 are the parameters to be estimated.

(b) For non-coached students (x2 = 0), the equation of the curve relating E(y) to x1 is:

E(y) = β0 + β1x1 + β3x1^2

The y-intercept is β0, which represents the expected SAT-Math score for a student with a PSAT score of 0 who did not receive coaching. The shift parameter is β1, which represents the expected change in SAT-Math score for a one-unit increase in PSAT score for non-coached students. The rate of curvature is β3, which represents the expected change in the slope of the curve for a one-unit increase in PSAT score.

(c) For coached students (x2 = 1), the equation of the curve relating E(y) to x1 is:

E(y) = β0 + β1x1 + β2 + β3x1^2 + β4

The y-intercept is β0 + β2, which represents the expected SAT-Math score for a student with a PSAT score of 0 who received coaching. The shift parameter is β1, which represents the expected change in SAT-Math score for a one-unit increase in PSAT score for coached students. The rate of curvature is β3, which represents the expected change in the slope of the curve for a one-unit increase in PSAT score. The term β4 represents the expected difference in the rate of change between coached and non-coached students.

(d) To test whether coaching has an effect on SAT-Math scores, we can conduct a hypothesis test on the coefficient of x2 (β2). The null hypothesis would be that β2 = 0, indicating that coaching has no effect on SAT-Math scores, while the alternative hypothesis would be that β2 ≠ 0, indicating that coaching does have an effect. We can use a t-test or an F-test to test this hypothesis, depending on the specific details of the regression analysis.